I have difficulty coming up with a practical example of a system such as described in the section “The Original Good Regulator Theorem”. Can you provide some examples?
Here are some of the problems I have in understanding the setup.
Minimising the entropy of Z says that Z is to have a narrow distribution, but says nothing about where the mean of that distribution should be. This does not look like anything that would be called “regulation”.
Time is absent from the system as described. Surely a “regulator” should keep the value of Z near constant over time?
The value of Z is assumed to be a deterministic function of S and R. Systems that operate over time are typically described by differential equations, and any instant state of S and R might coexist with any state of Z.
R has no way of knowing the value of Z. It is working in the dark. Why is it hobbled in this way?
Z is assumed to have no effect on S.
There are no unmodelled influences on Z. In practice, there are always unmodelled influences.
These problems still apply when the variables X and N are introduced.
Minimising the entropy of Z says that Z is to have a narrow distribution, but says nothing about where the mean of that distribution should be. This does not look like anything that would be called “regulation”.
I wouldn’t get too hung up on the word ‘regulator’. It’s used in a very loose way here, as in common in old cybernetics-flavoured papers. The regulator is regulating, in the the sense that it is controlling and restricting the range of outcomes.
Time is absent from the system as described. Surely a “regulator” should keep the value of Z near constant over time?
You could imagine that this is a memoryless system where the process repeats every timestep (and S/X has no memory of what Z was previously). Then, a regulator which chose a policy which minimized entropy of Z (and used the same policy each timestep) would keep the value of Z as near constant as possible. (Maybe this is closer to what you were thinking of as a ‘regulator’ the previous point?)
The value of Z is assumed to be a deterministic function of S and R. Systems that operate over time are typically described by differential equations, and any instant state of S and R might coexist with any state of Z.
This post deals with discrete systems where S and R happen ‘before’ Z. If you want something similar but for continuous systems over time, you might want to take a look at the Internal Model Principle, which is a similar kind of result, which can apply to continuous time systems modelled with differential equations.
R has no way of knowing the value of Z. It is working in the dark. Why is it hobbled in this way?
In this setup, S is just defined as the input to R. If you wanted, you could think of as S as the ‘initial state’ and Z as the ‘final state’ of the system, provided they had the same range. But this setup also allows S to have a range different to S. R can know the value of S/X, which is what it needs to know in order to affect Z. If it knows the dynamics of the setup then it can predict the outcome and doesn’t need to ‘see’ Z.
If you are thinking of something like ‘R must learn a strategy by trying out actions and observing their effect on Z’ then this is beyond the scope of this post! The Good Regulator Theorem(s) concern optimal behaviour, not how that behaviour is learned.
There are no unmodelled influences on Z. In practice, there are always unmodelled influences.
If by ‘unmodelled’ influences, you mean ‘variables that the regulator cannot see, but still affect Z’, then this problem is solved in the framework with X (and implicitly N).
I wouldn’t get too hung up on the word ‘regulator’. It’s used in a very loose way here, as in common in old cybernetics-flavoured papers.
Human slop (I’m referring to those old cybernetics papers rather than the present discussion) has no more to recommend it than AI slop. “Humans Who Are Not Concentrating Are Not General Intelligences”, and that applies not just to how they read but also how they write.
If you are thinking of something like ‘R must learn a strategy by trying out actions and observing their effect on Z’ then this is beyond the scope of this post! The Good Regulator Theorem(s) concern optimal behaviour, not how that behaviour is learned.
What I am thinking of (as always when this subject comes up) is control systems. A room thermostat actually regulates, not merely “regulates”, the temperature of a room, at whatever value the user has set, without modelling or learning anything. It, and all of control theory (including control systems that do model or adapt), fall outside the scope of the supposed Good Regulator Theorem. Hence my asking for a practical example of something that it does apply to.
Short Answer: It’s a toy model. I don’t think I can come up with a practical example which would address all of your issues.
Long Answer, which I think gets at what we disagree about:
I think we are approaching this from different angles. I am interested in the GRT from an agent foundations point of view, not because I want to make better thermostats. I’m sure that GRT is pretty useless for most practical applications of control theory! I read John Wentworth’s post where he suggested that the entropy-reduction problem may lead to embedded-agency problems. Turns out it doesn’t but it would have been cool if it did! I wanted to tie up that loose end from John’s post.
Why do I care about entropy reduction at all?
I’m interested in ‘optimization’, as it pertains to the agent-like structure problem, and optimization is closely related to entropy reduction, so this seemed like an interesting avenue to explore.
Reducing entropy can be thought of as one ‘component’ of utility maximization, so it’s interesting from that point of view.
Reducing entropy is often a necessary (but not sufficient) condition for achieving goals. A thermostat can achieve an average temperature of 25C by ensuring that the room temperature comes from a uniform distribution over all temperatures between 75C and −25C. But a better thermostat will ensure that the temperature is distributed over a narrower (lower entropy) distribution around 25C .
I think we probably agree that the Good Regulator Theorem could have a better name (the ‘Good Entropy-Reducer Theorem’?). But unfortunately, the result is most commonly known using the name ‘Good Regulator Theorem’. It seems to me that 55 years after the original paper was published, it is too late to try to re-brand.
I decided to use that name (along with the word ‘regulator’) so that readers would know which theorem this post is about. To avoid confusion, I made sure to be clear (right in the first few paragraphs) about the specific way that I was using the word ‘regulator’. This seems like a fine compromise to me.
I have difficulty coming up with a practical example of a system such as described in the section “The Original Good Regulator Theorem”. Can you provide some examples?
Here are some of the problems I have in understanding the setup.
Minimising the entropy of Z says that Z is to have a narrow distribution, but says nothing about where the mean of that distribution should be. This does not look like anything that would be called “regulation”.
Time is absent from the system as described. Surely a “regulator” should keep the value of Z near constant over time?
The value of Z is assumed to be a deterministic function of S and R. Systems that operate over time are typically described by differential equations, and any instant state of S and R might coexist with any state of Z.
R has no way of knowing the value of Z. It is working in the dark. Why is it hobbled in this way?
Z is assumed to have no effect on S.
There are no unmodelled influences on Z. In practice, there are always unmodelled influences.
These problems still apply when the variables X and N are introduced.
I wouldn’t get too hung up on the word ‘regulator’. It’s used in a very loose way here, as in common in old cybernetics-flavoured papers. The regulator is regulating, in the the sense that it is controlling and restricting the range of outcomes.
You could imagine that this is a memoryless system where the process repeats every timestep (and S/X has no memory of what Z was previously). Then, a regulator which chose a policy which minimized entropy of Z (and used the same policy each timestep) would keep the value of Z as near constant as possible. (Maybe this is closer to what you were thinking of as a ‘regulator’ the previous point?)
This post deals with discrete systems where S and R happen ‘before’ Z. If you want something similar but for continuous systems over time, you might want to take a look at the Internal Model Principle, which is a similar kind of result, which can apply to continuous time systems modelled with differential equations.
In this setup, S is just defined as the input to R. If you wanted, you could think of as S as the ‘initial state’ and Z as the ‘final state’ of the system, provided they had the same range. But this setup also allows S to have a range different to S. R can know the value of S/X, which is what it needs to know in order to affect Z. If it knows the dynamics of the setup then it can predict the outcome and doesn’t need to ‘see’ Z.
If you are thinking of something like ‘R must learn a strategy by trying out actions and observing their effect on Z’ then this is beyond the scope of this post! The Good Regulator Theorem(s) concern optimal behaviour, not how that behaviour is learned.
If by ‘unmodelled’ influences, you mean ‘variables that the regulator cannot see, but still affect Z’, then this problem is solved in the framework with X (and implicitly N).
Human slop (I’m referring to those old cybernetics papers rather than the present discussion) has no more to recommend it than AI slop. “Humans Who Are Not Concentrating Are Not General Intelligences”, and that applies not just to how they read but also how they write.
What I am thinking of (as always when this subject comes up) is control systems. A room thermostat actually regulates, not merely “regulates”, the temperature of a room, at whatever value the user has set, without modelling or learning anything. It, and all of control theory (including control systems that do model or adapt), fall outside the scope of the supposed Good Regulator Theorem. Hence my asking for a practical example of something that it does apply to.
Regarding your request for a practical example.
Short Answer: It’s a toy model. I don’t think I can come up with a practical example which would address all of your issues.
Long Answer, which I think gets at what we disagree about:
I think we are approaching this from different angles. I am interested in the GRT from an agent foundations point of view, not because I want to make better thermostats. I’m sure that GRT is pretty useless for most practical applications of control theory! I read John Wentworth’s post where he suggested that the entropy-reduction problem may lead to embedded-agency problems. Turns out it doesn’t but it would have been cool if it did! I wanted to tie up that loose end from John’s post.
Why do I care about entropy reduction at all?
I’m interested in ‘optimization’, as it pertains to the agent-like structure problem, and optimization is closely related to entropy reduction, so this seemed like an interesting avenue to explore.
Reducing entropy can be thought of as one ‘component’ of utility maximization, so it’s interesting from that point of view.
Reducing entropy is often a necessary (but not sufficient) condition for achieving goals. A thermostat can achieve an average temperature of 25C by ensuring that the room temperature comes from a uniform distribution over all temperatures between 75C and −25C. But a better thermostat will ensure that the temperature is distributed over a narrower (lower entropy) distribution around 25C .
I think we probably agree that the Good Regulator Theorem could have a better name (the ‘Good Entropy-Reducer Theorem’?). But unfortunately, the result is most commonly known using the name ‘Good Regulator Theorem’. It seems to me that 55 years after the original paper was published, it is too late to try to re-brand.
I decided to use that name (along with the word ‘regulator’) so that readers would know which theorem this post is about. To avoid confusion, I made sure to be clear (right in the first few paragraphs) about the specific way that I was using the word ‘regulator’. This seems like a fine compromise to me.